So, what's to stop someone like me from taking a green laser module, and replacing the 808nm diode in it with a higher powered 808nm diode? Or, is there a way to get the same type(s) of crystals used in a green laser, and just put the whole thing together myself?

The crystals in pen lasers are pretty much the same size so in theory you could buy a 5mw green and swop out the diode for a higher powered one. StoneTek has some nice 200mW 808's which should fit the crystal/len module of the pen although the final output would be about half the diode power, 5mW to 100mW is still a tasty leap

There are alot of posts about showing you how to replace the IR diode of green pens with info on removing the cap and aligning it.

Also, for an increase of that size you'll need to change the driver. I doubt the one in the pen could be adjusted that high.